23 11
Article 07.8.4
Journal of Integer Sequences, Vol. 10 (2007), 2
3 6 1 47
Exponential Generating Functions for Trees
with Weighted Edges and Labeled Nodes
Wen-jin Woan
1Howard University
Washington, DC 20059
USA
wjwoan@sbcglobal.net
Barbara Tankersley
North Carolina Agricultural and Technical State University
Greensboro, NC 27411
USA
tankers@ncat.edu
Abstract
We study labeled trees and find that this is a class of combinatorial objects which is perfect for the study of some exponential generating functions.
1
Introduction
In this paper we study the counting of various trees with labeled nodes and weighted edges and its exponential generating functions. Section 1 is about rooted plane trees. Section 2 is about labeled trees. In Section 3, we give some examples of labeled trees and its exponential generating functions. In Section 4, we discuss weighted and labeled trees and its exponen-tial generating functions. In Chapter 1 of [4], two tree representations of permutations are studied, here we study the combinatorics of the second tree representation through gener-ating functions and show the connections with some classical sequences. A similar study corresponding to the first tree representation of [4] has been done in [2,7].
The counting of trees (rooted plane trees) is the Catalan sequence,c0 = 1,1,2,5,14,42, . . .
Traverse the tree starting from the root going up on the left side of the branches and coming
down on the right side of the branches. On the way down if we encounter a right branch we go up. Label the nodes in order with [n] = {0,1,2,3, . . . , n} by preorder, i.e., label the nodes at the first visit.
Example 1. Rooted plane trees of order n= 3, c3 = 5
❅ ❅❅
❅ ❅
❅ ❅❅
❅
❅ ❅ ❅
❧ ❧ ❧ ❧
❧ ❧ ❧
❧ ❧ ❧
❧ ❧ ❧
❧ ❧
❧
❧
❧ ❧ ❧
0 1 2 3
0 1
2 3
0 1 2
3
0
1 2
3
0 1 2 3
2
Labeled Trees
In this section we define labeled trees and study some well known partitions of permutations. LetP =a1a2a3· · ·an be a permutation, locate the smallest numberai.Partition P =F aiB byF the part before ai and B the part after ai, then inductively label the first node of the left most branch by ai and put the subtree for F on top of ai and the subtree for B to the right ofai.Note that the labeled tree satisfies the following conditions:
(1) Increasing on any path going up,
(2) Increasing from left to right of the immediate siblings of a node.
Definition 2. A labeled tree is a tree with nodes labeled by [n] = {0,1,2,3,4, . . . , n} which satisfies the above two conditions. It is also called an increasing tree, please refer to [4].
Example 3. We read the labeled trees in postorder and they correspond to permutations as follows: Forn = 3, the third tree has two choices (2 or 3) for position 2 and the count is
s3 = 1 + 1 + 2 + 1 + 1 = 6.
Permutations: 321,231,213,312,132,123.
❅ ❅
❅
❅ ❅❅ ❅❅ ❅❅
♠ ♠ ♠ ♠
♠ ♠
♠ ♠
♠
♠ ♠
♠
♠
♠ ♠
♠
♠
♠ ♠
♠
♠ ♠ ♠ ♠
0 1 2 3
0 1
2 3
0 1 2
3
0 1 3
2
0
1 2
3
0 1 2 3
3
Some Examples of Labeled Trees
In this section we study some examples of labeled trees and its exponential generating func-tions(EGF). Let A(x) = P 1
n!anx
n and B(x) = P 1
n!bnx
n be the exponential generating
functions for the sequences{an}, {bn}, then A(x)B(x) =P 1
n!(
P n k
akbn−k)x
n (1)
On the other hand, if we let tn be the number of labeled trees and partition the tree by the first node of the leftmost branch, then we have the recurrence
tn = Pn−1
k=0
n−1
k
tktn−1−k. (2)
We choose k numbers from {2,3,4, . . . , n} to form a subtree on top of node 1 and the rest to the right of node 1.
By the similarity of (1) and (2), it certainly is appropriate to study its exponential generating functions instead of ordinary generating functions.
Remark 4. A tree is said to be a 1-2 tree, if every vertex has outdegree at most two. The counting of 1-2 trees is the Motzkin sequence, 1,1,2,4,9,21,51, . . .. The counting of labeled 1-2 trees is the sequence of zig-zag permutations, t0 = 1,1,2,5,16,61,272, . . .
The following are the list of labeled 1-2 trees for n = 3,4
❅ ❅
❅
❅ ❅❅ ❅❅
♠ ♠ ♠ ♠
♠ ♠
♠ ♠
♠
♠ ♠
♠
♠
♠ ♠
♠
♠
♠ ♠
♠
0 1 2 3
0 1
2 3
0 1 2
3
0 1 3
2
0
1 2
3
❅
exponential generating function of the sequence of zig-zag permutations.
Proof. Partition the trees by the out degrees of the root. We have the recurrence relation bn=bn−1+
Thus y=f(x) satisfies the differential equation dy
dx),since we need the coefficient of [xn
n!], so
R
ydydxdx= y2
2 .
Solving the above differential equation, yields f(x) = y= tan2x+π
Example 6. The counting of the complete binary tree is 1,0,1,0,2,0,5,0, . . . and the counting of labeled complete binary tree is 1,0,1,0,4,0,34,0,496, . . .
Let bi be the number of labeled complete binary trees and the exponential generating function
y=f(x) = P i=0bi x
i+1
(i+1)!, then
dy
dx = 1 +
1 2!y
2.
Solve the differential equation y= √2 tan x
√
2 =x+ 1 6x
3+ 1 30x
5+ 17 2520x
7+ 31 22680x
9+O(x11).
Example 7. The counting of the labeled complete ternary trees is the sequence b0 =
1,0,0,1,0,0,15,0,0,855, . . .Its exponential generating function satisfies the differential equa-tion
dy
dx = 1 + y3
3!.
Solving the differential equation and express x in terms ofy x= 1
3(
3
√
6 ln(y+√3
6)−√36 2 ln(y
2−√3
6y+(√3
6)2)+√3(√3
6)(arctan( 2
√
3√36y− 1
√
3))+ 1 6
√
3√3
6π) =y− 241y4+ 2521 y7− 21601 y10+168481 y13− 1244161 y16+O(y19).
Using the Lagrange Inversion Formula, we find y=x+ 1
4!x 4+ 15
7!x 7+ 855
10!x
10+121605 13! x
13+. . .
Example 8. We define a Bell tree to be a 1-2 tree with the right branch a single file. For labeled Bell trees, the counting is the Bell number sequence 1,1,2,5,15,52,203, . . .
The recurrence relation is an =P n−k1
ak∗(1) and its exponential generating function y=P
an(n+1)!1 xn+1 satisfies the differential equation dy
dx =ye x.
Solving the differential equation we have y= exp(exp(x)−1) = 1 +x+x2+ 5
6x 3+5
8x 4+ 13
30x 5+ 203
720x
6+ 877 5040x
7+O(x8).
For n= 4, there are 16 labeled 1-2 trees. The only labeled 1-2 tree that is not counted as Bell trees is the last one in Remark4.
Example 9. Trees of height at most two. The recurrence relation is the same as in the previous example.
Example 10. Trees of height at most three. The recurrence relation is bn=
P n−1
k
ak∗bn−1−k,where ak is the counting of trees of height at most two.
y′ = yf,where f is EGF of {an}as in the previous example. ln y=R
f =R
exp(exp(x)−1), y= exp(R
(exp(exp(x)−1))dx) = 1 +x+x2+x3+ 23 24x4+
53
60x5+O(x6).
4
Weighted Labeled 1-2 Trees
of the weights of all edges. Let W Tn(b, c) be the set of all labeled and weighted increasing trees of order n with weights b, c and wn be the total weight of all trees in W Tn(b, c), then we have the following.
Theorem 11. Let y =f(x) =P wn
(n+1)!x
n+1 be the exponential generating function of {w
n}. Then it satisfies the following differential equation
y′
The solution is as follows: Case 1. 2c−b2 = 0. y = −b
Proof. By similar idea in Theorem 5 we can obtain the differential equation, then solve the separable differential equation.
Example 12. Labeled 1-2 trees with weightsb = 3,c= 4, the sequence is 1,1,3,13,75,541, . . ., (A000670). The sequence counts the number of ways n competitors can rank in a competi-tion, allowing for the possibility of ties.
y′ = 1 + 3y+ 2!4y2, (A080635). It counts the number of permutations onnletters without double falls and with-out initial falls.
y′ = 1 +y+y2, (A050351). It counts the number of 3-level labeled linear rooted trees with n leaves.
y′
[1] F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-like Structures, Cambridge University Press, 1998.
[3] N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, http://www.research.att.com/~ njas/sequences.
[4] R. P. Stanley,Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.
[5] R. P. Stanley,Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999.
[6] H. S. Wilf,Generatingfunctionology, Academic Press, 1994.
[7] J. Zeng, Enum´eration de permutations et J-fractions continues, European J. Combin., 14 (1993), 373–382.
2000 Mathematics Subject Classification: Primary 05A15.
Keywords: labeled tree, Motzkin path, Catalan path, weighted labeled tree, exponential generating function.
(Concerned with sequences A000670, A000828,A002866, A050351, and A080635.)
Received May 14 2007; revised version received August 7 2007. Published in Journal of Integer Sequences, August 13 2007.
